harmonics

A sonometer. The top wire is sounding its fundamental
note, the others are sounding their 2nd, 3rd, 4th, and 5th harmonics.

Vibrations at frequencies which are integer
multiples of that of a fundamental vibration: the ascending notes C, G,
E, C', E', G' comprise a fundamental with its first five higher harmonics.
Apart from their musical consonance, they are important because any periodically
repeated signal – a vowel sound, for example – can be produced
by superposing the harmonics of the fundamental frequency, each with the
appropriate intensity and time lag.

Introduction

Music would be unimaginably dull if all musical instruments playing a certain
note – say middle C – were to sound exactly alike. A trumpet
and a violin would produce indistinguishable sounds and there would be no
point in making different musical instruments. Both of these instruments
are capable of producing the same note and yet sound completely different.
They make the air around them vibrate with a certain frequency. In the case
of middle C, this is at 256 vibrations per second. The basic vibration with
the largest possible wave size is called the fundamental. The best known
instrument capable of producing the fundamental without producing additional
waves is the tuning fork (which is used for tuning pianos) and because of
this gives out a very "tinny" note. Additional sound waves give the body
or quality to musical notes. These additional sound waves are called harmonics.

When a note is played, fundamental vibrations are set up and also harmonics
or overtones. A vibration with a wave half
as long as the fundamental is called the second harmonic; a vibration with
a wave one-third as long is called the third harmonic and so on. The different
sound qualities of instruments depend on the fact that together with the
fundamental, or first harmonic, certain other harmonics are played, some
more loudly than others.

This sonometer has two identical wires. A bridge
is placed one third the way along the back wire thereby stopping the
wire from moving and forming a node. The shorter section of this wire
is plucked in the middle. The front wire vibrates in sympathy. Pieces
of paper jump off this wire at antinodes (maximum vibration) and stay
on at nodes (no vibration).

Wave forms on an oscilloscope

Pictures of the wave forms of sounds can be obtained by using an electronic
apparatus called the cathode ray oscilloscope. It gives a still picture
of wave forms that really are repeating hundreds or thousands of times a
second. The picture will be of a very complicated wave which is a combination
of the fundamental and various harmonics and will need to be unscrambled
by a mechanical harmonics analyzer into simpler waves, one representing
each harmonic.

The waves show how the complicated wave pattern of
the G note of the clarinet is built up by combining the fundamental
with its various harmonics. The shape of the final pattern is largely
determined here by the addition of the fundamental and third harmonics
(the second harmonic is absent). The 4th and 5th harmonics only slightly
alter the wave shape.

Harmonics and sound quality

The harmonics of some waves are so faint that they can be ignored. In the
case of a clarinet, playing the note G above middle C, the first and third
harmonics are the most important, the fourth and fifth are of lesser importance.
The second harmonic is almost undetectable.

Two different types of organ pipes playing the same note produce different
sounds due to the presence of different harmonics or overtones. The two
types of organ pipes are the open organ pipe which has both ends open and
produces antinodes (places where vibrations are largest) at each end, and
the closed organ pipe which produces an antinode at the open end and a node
(place of no vibration) at the closed end.

Open organ pipes (i) always have antinodes (where
the air molecules vibrate most vigorously) at their open ends and
can make all the possible harmonics (2nd, 3rd, 4th, etc.), whereas
closed ones (ii) which must have nodes (where the air molecules are
at rest) at their closed ends can only give odd-numbered harmonics
(3rd, 5th, etc.)

Diagram (i) shows how the air in an open organ pipe can vibrate in a number
of ways, in each case keeping an antinode at each end. The simplest method
of vibration occurs when there is just one node, half way along the pipe.
The sound produced by this vibration is the fundamental or first harmonic.
At the same time the air in the pipe can vibrate so that there are two nodes,
one-quarter and three-quarters the way along the pipe. This gives rise to
the second harmonic – a sound whose wavelength is half that of the
fundamental. The same pipe can also produce third, fourth, fifth, and higher
harmonics.

Diagram (ii) shows how the air in an organ pipe closed at one end can vibrate
in a number of ways, in each case keeping a node at the closed end and an
antinode at the open end. With just one node and one antinode and pipe sounds
in the fundamental note or first harmonic. With a second node one-third
of the way up the pipe the sound produced has a wavelength of one-third
that of the fundamental. This gives to the third harmonic. A closed pipe
has no second harmonic – it has only odd-numbered harmonics. Because
it lacks even-numbered harmonics the closed organ pipe gives a sound of
different quality from the open-ended pipe, though their fundamental notes
may be the same.

Harmonics or overtones are what makes the violin sound different from the
euphonium. They give the notes their distinctive quality.

Harmonics. (A) The first 16 modes of harmonic vibration
of a stretched string. (B) The musical pitches these would sound if
the fundamental (first harmonic) sounded C. (C) The ratios between
the frequencies of the different modes of vibration. Thus, the frequency
of the fundamental is half (1:2) that of the first overtone (second
harmonic) and one-third that of the second overtone (third harmonic);
that of the second overtone is two-thirds that of the third overtone,
and so on. Note that after the fundamental (the first harmonic), the
nth overtone is the (n + 1)th harmonic.